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Full text of "Mathematical And Physical Papers - Iii"

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conventional abbreviations, we must say that the intensity is measured, not by c2, but by the mean value of c2, which may conveniently be represented by ttt(c2).
2. Let us examine now whether it be always possible to resolve the given disturbance into two which, taken separately, would correspond to two elliptically polarized streams of given nature. For the sake of clear ideas, it may be supposed that the azimuths and eccentricities of the ellipses belonging to these two streams are given and invariable, while the azimuth and eccentricity of the ellipse belonging to the first stream are given for that stream, but vary from one to another of a set of streams which we wish to consider in succession.
Let #1? c1? &c. be for the first, and #2, c2, &c. be for the second stream of the pair, what CD', c, &c. were for the original stream ; and resolve all the displacements along the principal axes of the latter stream. Then, in order that the original disturbance may be equivalent to the pair, we must have, independently of 0,
cos (Oj -)-#! sin (oj - a) + #2 cos (2 - a) xl sin (ax  a) + yl cos (ax - a) + #2 sin (a2 - a)
Conceive xl, y1, #2, ?/2, x', and y' expressed in terms of $ by the formulae (2) and the similar formula whereby x1} ylt &c. are expressed, and then let the sines and cosines of <f> + e, 0 + ^, and 0H-e2 be developed. In order that equations (3) may be satisfied independently of <, the coefficients of sin </> and cos </> must separately be equal to zero, so that each of these equations will split into two. We shall thus have four equations to determine the four unknown quantities c1? c2, e,, and e2. For the sake of shortness, let
c cos e = gt                       c sin e  h,
whether the letters be or be not affected with suffixes; and further put